Let $u\in H^1_{\mathrm{loc}}(\Omega)$ weakly solve
$$
D_i(a^{ij}D_ju)=D_iF^i+g,
$$
where $a^{ij},F^i\in C^{0,\alpha}$, $g\in L^p$ for a suitable $p>n/(1-\alpha)$, and $a^{ij}$ is uniformly elliptic. The interior divergence-form Schauder estimate states that for $\Omega'\Subset\Omega_1\Subset\Omega$,
$$
\boxed{\|u\|_{C^{1,\alpha}(\Omega')}
\leq C\left(\|u\|_{L^2(\Omega_1)}
+\|F\|_{C^{0,\alpha}(\Omega_1)}+\|g\|_{L^p(\Omega_1)}\right).}
$$
For the homogeneous equation the final two terms vanish.


Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 107 / 2 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-107/2/d/solution